Paddy McCrudden - Representations of quantum categories

A quantum monoid in a symmetric monoidal category , is a monoid in with extra structure which ensures that the category of representations is a braided monoidal category with a twist. In the case that is the category of vector spaces, a quantum monoid is a quasi-triangular quasi-bialgebra with a twist and a quantum group is a quantum monoid with an antipode.

I will prove a converse to the first sentence: that is, if M is a monoid in and the category of representations of M is a braided monoidal object with a twist (in a certain symmetric monoidal bicategory) then M is a quantum monoid. I will also extend this theorem to the several object case; quantum categories.

To prove this I will exhibit a biequivalence of symmetric monoidal bicategories which induces a duality between quantum categories and their categories of representations.